Prof. Dr. M. Wolf WS 2018/19
M. Heinze Sheet 8
Differential Topology: Exercise Sheet 8
Exercises (for Feb. 6th and 7th)
8.1 Definef, g :R3 :→R as
f(x, y, z) =x3y2z2+y4x−7xz2 (1) g(x, y, z) = 2x3z8−5y2z+yx4 . (2) Prove, using differential topology, that f and g have a common non-zero zero, i.e that there is an a∈R3\ {0} such thatf(a) =g(a) = 0.
8.2 Letf :Sn→Sn be a smooth map that carries the antipodal points to antipodal points.
Compute deg2(f).
8.3 Construct a diffeomorphism f : Sn → Sn for which you prove that it is not smoothly homotopic to the identity map.
8.4 For any n∈N provide a smooth map f :S1 →S1 with deg(f) =n.
8.5 Let f :Sn →Sn be smooth and n even. Compute maxx∈Sn|hf(x), xi| where x and f(x) are regarded as unit vectors in Rn+1.
8.6 Consider the following system of equations
2x+y+ sin(x+y) = 0 (3)
x−2y+ cos(x+y) = 0 . (4)
Use the Euclidean degree to prove or disprove that there is a solution (x0, y0)∈R2 with x20+y02 < 14.